Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the curves about the given axis. y = 3e^x, y = 3e^(−x), x = 1; about the y-axis

Answer 1

To start it off, we have to find where the 3 meet - to find when 3e^x and 3e^(-x) meet, we start off with dividing both sides to get e^x=e^(-x) and e^x=1/e^x, multiplying both sides by e^x to get e^(2x)=1 and x=0. Plugging in x=1 for both equations, we get 3e and 3/e respectively. Graphing it out, we can see that it forms a weird shape, but if you draw a line at y=3, we can have 2 separate shapes, making it super easy! We have x as the radius since it's about the y axis and the equations (from a certain point) as the height
We have 2 integrals - 2π(∫(from 3 to 3e) (x)(3e^x)) and  2π(∫(from 3/e to 3) (x)(e^(-x)). We then get 2π (∫3xe^x+∫3xe^(-x)) as added up they make the area between the curves.For ∫3xe^x,  For 3xe^x, which we can separate the 3 from and get xe^x, we use integration by parts to put x in for f and e^x as g in ∫fg'=fg-∫f'g, plugging it in to get xe^x-∫e^x, resulting in xe^x-e^x. Multiplying that by 3 (as we separated that earlier), we get 3xe^x-e^x 
For ∫3xe^(-x), we can use the same technique for separating the 3 out, but for ∫xe^(-x), we can use put x in for f and g' as e^(-x), resulting in g being -e^(-x) by using u substitution and making -x u. Next, we get (x)(-e^(-x))+∫(-e)^(-x), and ∫(-e)^(-x)=e^(-x) in the same way -e^(-x) being the integral of e^(-x) was found.
Adding it all up for this, we get 3(-xe^(-x)-e^(-x)) as our solved integral. 

Since 3xe^x-3e^x is just the solved integral and we need to find it on 3 to 3e, we plug in 3e for x and subtract it from plugging 3 into x to get ((9*e-3)*e^(3*e)-6*e^3). We need to multiply it by 2pi (since it's a cylindrical shell) to get (2pi(((9*e-3)*e^(3*e)-6*e^3))

For 3(-xe^(-x)-e^(-x)) , we can plug it in for 3 and 3/e (as the respective upper and lower bounds) to get 3*((e+3)*e^(-3*e^(-1)-1)-4*e^(-3)). Multiply that by 2pi to get 6pi*((e+3)*e^(-3*e^(-1)-1)-4*e^(-3)).
Add the two up to get 6pi*((e+3)*e^(-3*e^(-1)-1)-4*e^(-3))+(2pi(((9*e-3)*e^(3*e)-6*e^3)) in and it said